Understanding and monitoring the complex and dynamic processes of the Sun is important for a number of human activities on Earth and in space. For this reason, NASA's Solar Dynamics Observatory (SDO) has been continuously monitoring the multi-layered Sun's atmosphere in high-resolution since its launch in 2010, generating terabytes of observational data every day. The synergy between machine learning and this enormous amount of data has the potential, still largely unexploited, to advance our understanding of the Sun and extend the capabilities of heliophysics missions. In the present work, we show that deep learning applied to SDO data can be successfully used to create a high-fidelity virtual telescope that generates synthetic observations of the solar corona by image translation. Towards this end we developed a deep neural network, structured as an encoder-decoder with skip connections (U-Net), that reconstructs the Sun's image of one instrument channel given temporally aligned images in three other channels. The approach we present has the potential to reduce the telemetry needs of SDO, enhance the capabilities of missions that have less observing channels, and transform the concept development of future missions.